Applying Logo to Sphere to move to Unity
in Edit mode select all vertices and Unwrap as Cube Projection
go to UV Editor and Scale the UV a bit up
go to Shader Editor and for Image node - change Repeat to Clip


Note: in this case I didn't use Texture Coordinates node since UV is used by default and Mapping node since positioning is controlled by UV Map
Unity
Im not experienced with Unity, so there is probably better way, but looks like

in Blender - Bake material into textures (Color (Albedo), Metalic, Normal, Ambient,  ... etc for PBR material)
export as FBX
in Unity import asset and assign baked texture to Albedo channel


Check documentation https://docs.unity3d.com/Manual/Materials.html
Also search Blender to Unity topic like here https://blender.stackexchange.com/a/132340/2214